Observation of Time-Invariant Coherence in a Nuclear Magnetic Resonance Quantum Simulator

Isabela A. Silva,1,2 Alexandre M. Souza,3 Thomas R. Bromley,2 Marco Cianciaruso,2,4

Raimund Marx,5 Roberto S. Sarthour,3 Ivan S. Oliveira,3 Rosario Lo Franco,1,2,6,7

Steffen J. Glaser,5 Eduardo R. deAzevedo,1 Diogo O. Soares-Pinto,1 and Gerardo Adesso2,∗

1Instituto de Fısica de Sao Carlos, Universidade de Sao Paulo, CP 369, 13560-970, Sao Carlos, SP, Brazil2School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

3Centro Brasileiro de Pesquisas Fısicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil4Dipartimento di Fisica “E. R. Caianiello”, Universita degli Studi di Salerno, Via Giovanni Paolo II, I-84084 Fisciano (SA), Italy

5Department of Chemistry, Technische Universitat Munchen, Lichtenbergstr. 4, 85747, Garching, Germany6Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici,

Universita di Palermo, Viale delle Scienze, Edificio 9, 90128 Palermo, Italy7Dipartimento di Fisica e Chimica, Universita di Palermo, Via Archirafi 36, 90123 Palermo, Italy∗To whom correspondence should be addressed; E-mail: gerardo.adesso@nottingham.ac.uk

(Dated: September 19, 2016)

The ability to live in coherent superpositions is a signature trait of quantum systems and constitutes an irre-placeable resource for quantum-enhanced technologies. However, decoherence effects usually destroy quantumsuperpositions. It has been recently predicted that, in a composite quantum system exposed to dephasing noise,quantum coherence in a transversal reference basis can stay protected for indefinite time. This can occur for aclass of quantum states independently of the measure used to quantify coherence, and requires no control onthe system during the dynamics. Here, such an invariant coherence phenomenon is observed experimentally intwo different setups based on nuclear magnetic resonance at room temperature, realising an effective quantumsimulator of two- and four-qubit spin systems. Our study further reveals a novel interplay between coherenceand various forms of correlations, and highlights the natural resilience of quantum effects in complex systems.

Successfully harnessing genuine nonclassical effects is pre-dicted to herald a new wave of technological devices with adisruptive potential to supersede their conventional counter-parts [1]. This prediction is now coming of age, and an inter-national race is on to translate the power of quantum technolo-gies into commercial applications to networked communica-tion, computing, imaging, sensing and simulation [2]. Quan-tum coherence [3], which incarnates the wavelike nature ofmatter and the essence of quantum parallelism [4], is the pri-mary ingredient enabling a supraclassical performance in awide range of such applications. Its key role in quantum al-gorithms, optics, metrology, condensed matter physics, andnanoscale thermodynamics is actively investigated and widelyrecognised [5–13]. Furthermore, coherent quantum effectshave been observed in large molecules [14] and are advo-cated to play a functional role in even larger biological com-plexes [15–18]. However, coherence is an intrinsically frag-ile property which typically vanishes at macroscopic scales ofspace, time, and temperature: the disappearance of coherence,i.e. decoherence [19], in quantum systems exposed to environ-mental noise is one of the major hindrances still threateningthe scalability of most quantum machines. Numerous effortshave been thus invested in recent years into devising feasi-ble control schemes to preserve coherence in open quantumsystems [20], with notable examples including dynamical de-coupling [21, 22], quantum feedback control [23] and errorcorrecting codes [24].

In this Letter we demonstrate a fundamentally differentmechanism. We observe experimentally that quantum coher-ence in a composite system, whose subsystems are all affectedby decoherence, can remain de facto invariant for arbitrarilylong time without any external control. This phenomenon wasrecently predicted to occur for a particular family of initialstates of quantum systems of any (however large) even num-

ber of qubits [25], and is here demonstrated in a room temper-ature liquid-state nuclear magnetic resonance (NMR) quan-tum simulator [29–33] with two different molecules, encom-passing two-qubit and four-qubit spin ensembles. After ini-tialisation into a so-called generalised Bell diagonal state [25],the multiqubit ensemble is left to evolve under naturally oc-curring phase damping noise. Constant coherence in a refer-ence basis (transversal to the noise direction) is then observedwithin the experimentally considered timescales up to the or-der of a second. Coherence is measured according to a vari-ety of recently proposed quantifiers [26], and its permanenceis verified to be measure-independent. We also reveal howcoherence captures quantitatively a dynamical interplay be-tween classical and general quantum correlations [34], whileany entanglement may rapidly disappear [35]. For more gen-eral initial states, we prove theoretically that coherence candecay yet remains above a guaranteed threshold at any time,and we observe this experimentally in the two-qubit instance.The present study advances our physical understanding of theresilience of quantum effects against decoherence.

Quantum coherence manifests when a quantum system is ina superposition of multiple states taken from a reference ba-sis. The reference basis can be indicated by the physics of theproblem under investigation (e.g. one may focus on the energyeigenbasis when addressing coherence in transport phenom-ena and thermodynamics) or by a task for which coherenceis required (e.g. the estimation of a magnetic field in a cer-tain direction). Here, for an N-qubit system, having in minda magnetometry setting [36], we can fix the reference basis tobe the ‘plus/minus’ basis {|±〉⊗N}, where {|±〉} are the eigen-states of the σ1 Pauli operator, which describes the x compo-nent of the spin on each qubit [24]. Any state with densitymatrix δ diagonal in the plus/minus basis will be referred toas incoherent. According to a recently formulated resource

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FIG. 1. A: Pulse sequence to prepare two-qubit BDstates encoded in the 1H and 13C nuclear spins of Chlo-roform; the rf pulse (θ)µ realises a qubit rotation by θabout the spin-µ axis, J is the scalar spin-spin cou-pling, and time flows from left to right. B: Dynam-ics of the experimental states ρ(t) (magenta points)in the space of the spin-spin correlation triple c j =

〈σ j ⊗ σ j〉, j = 1, 2, 3; all BD states (2) fill the lightblue tetrahedron, while the subclass of states spanningthe inscribed green surface are predicted to have time-invariant coherence in the plus/minus basis accordingto any measure of Eq. (1) [25]. C–F: full tomogra-phies of the experimental states as prepared at timet = 0 (C, D) and after t = 0.25 s of free evolution(E, F), recorded in the computational basis (top row)and in the plus/minus basis (bottom row). G: Evolu-tion of (absolute values of) the correlation functions|c j | in the experimental states (points), along with the-oretical predictions (solid lines) based on phase damp-ing noise with our measured relaxation times. H: Ex-perimental observation of time-invariant coherence (inthe plus/minus basis), measured by relative entropy(red circles) [26], fidelity-based measure (blue trian-gles) [27], and normalised trace distance (green dia-monds) [25], equal to l1 norm [26] in BD states. Theslight negative slope is due to the subdominant effectof amplitude damping. I: Experimental dynamics ofcoherence and all forms of correlations [28] measuredvia relative entropies (theoretical curves omitted forgraphical clarity). In panels G–I, experimental errorsdue to small pulse imperfections (0.3% per pulse) re-sult in error bars within the size of the data points.

theory [3, 26, 27, 37], the degree of quantum coherence in thestate ρ of a quantum system can be quantified in terms of howdistinguishable ρ is from the set of incoherent states,

CD(ρ) = infδ incoherent

D(ρ, δ) , (1)

where the distance D is assumed jointly convex and contrac-tive under quantum channels, as detailed in the SupplementalMaterial [38]. In general, different measures of coherence in-duce different orderings on the space of quantum states, as ithappens e.g. for entanglement or other resources. A conse-quence of this is that, for states of a single qubit, it is impossi-ble to find a nontrivial noisy dynamics under which coherenceis naturally preserved when measured with respect to all possi-ble choices of D in Eq. (1). As predicted in [25], such a coun-terintuitive situation can occur instead for larger compositesystems. Here we observe this phenomenon experimentally.

Our NMR setup realises an effective quantum simulator,in which N-qubit states ρ can be prepared by manipulatingthe deviation matrix from the thermal equilibrium densityoperator of a spin ensemble [29, 31], via application of ra-diofrequency (rf) pulses and evolution under spin interactions[24, 33]. The scalability of the setup relies on availability ofsuitably large controllable molecules in liquid-state solutions.

We first encoded a two-qubit system in a Chloroform(CHCl3) sample enriched with 13C, where the 1H and 13Cspin- 1

2 nuclei are associated to the first and second qubit, re-spectively. This experiment was performed in a Varian 500MHz liquid-NMR spectrometer at room temperature, accord-ing to the plan illustrated in Fig. 1A. The state preparationstage allowed us to initialise the system in any state obtainedas a mixture of maximally entangled Bell states, that is, any

Bell diagonal (BD) state [47]. These states take the form

ρ = 14

(I ⊗ I +

∑3j=1 c j σ j ⊗ σ j

), (2)

where {σ j} are the Pauli matrices and I is the identity operatoron each qubit; they are completely specified by the spin-spincorrelation functions c j = 〈σ j ⊗σ j〉 for j = 1, 2, 3, and can beconveniently represented in the space spanned by these threeparameters as depicted in Fig. 1B. We aimed to prepare specif-ically a BD state with initial correlation functions c1(0) = 1,c2(0) = 0.7 and c3(0) = −0.7, by first initialising the system inthe pseudo-pure state |00〉〈00| as described in Refs. [24, 29],and then implementing the sequence of rf pulses shown inFig. 1A with θ = π and α = arccos(−0.7) ≈ 134◦.

After state preparation, the system was allowed to evolvefreely during a period of time t, with t increased for each trialin increments of 2/J from 0 to 0.5 s (where J ≈ 215 Hz isthe scalar spin-spin coupling constant [38]), in order to ob-tain the complete dynamics. In the employed setup, the twomain sources of decoherence can be modelled as Markovianphase damping and generalised amplitude damping channelsacting on each qubit, with characteristic relaxation times T2and T1, respectively [38]. For our system, the relaxation timeswere measured as T H

1 = 7.53 s, T H2 = 0.14 s, TC

1 = 12.46s, TC

2 = 0.90 s which implies that T H,C1 � T H,C

2 . Therefore,considering also the time domain of the experiment, only thephase damping noise can be seen to have a dominant effect.

The final stage consisted of performing full quantum statetomography for each interval of time t, following the proce-dure detailed in [38, 48]. Instances of the reconstructed ex-perimental states at t = 0 and t = 0.25 s are presented inFig. 1C–F. The fidelity of the initial state with the ideal target

3

was measured at 99.1%, testifying the high degree of accuracyof our preparation stage. We verified that the evolved state re-mained of the BD form (2) during the whole dynamics withfidelities above 98.5%: we could then conveniently visualisethe dynamics focusing on the evolution of the spin-spin corre-lation triple {c j(t)}, as indicated by magenta points in Fig. 1B.The time evolution of the triple {c j(t)} is detailed in Fig. 1G.

From the acquired state tomographies during the relaxationprogress, we measured the dynamics of quantum coherence inour states adopting all the known geometric coherence mono-tones proposed in the literature, as shown in Fig. 1H. All quan-tifiers were found simultaneously constant within the experi-mental confidence levels, revealing a universal resilience ofquantum coherence in the dynamics under investigation. Notethat the observed time-invariant coherence is a nontrivial fea-ture which only occurs under particular dynamical conditions.A theoretical analysis [25, 38] predicts in fact that, for all BDstates evolving such that their spin-spin correlations obey thecondition c2(t) = −c1(t)c3(t) (corresponding to the lime greensurface in Fig. 1B), any valid measure of coherence as de-fined in Eq. (1) with respect to the plus/minus basis shouldremain constant at any time t. As evident from the placementof the data points in Fig. 1B, our setup realised precisely thepredicted dynamical conditions for time-invariant coherence,with no further control during the relaxation. Our experi-ment thus demonstrated a nontrivial spontaneous occurrenceof long-lived quantum coherence under Markovian dynamics.

We remark that the observed effect is distinct from the phys-ical mechanism of long-lived singlet states also studied inNMR [49], and from an instance of decoherence-free sub-space [50]. In the latter case, an open system dynamics canact effectively as a unitary evolution on a subset of quan-

| 01⟩⟨

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y

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FIG. 2. A: Modified preparation stage to engineer non-BD two-qubit states.We prepared two states ρ1 and ρ2 with purity 0.92 and 0.93 respectively, set-ting phases θ ≈ 0.94 rad, α = π/3 for ρ1, and θ ≈ 0.78 rad, α = π/2 for ρ2.The evolution and acquisition stages were as in Fig. 1A. Full tomographies ofthe produced states at t = 0 are presented in: C (ρ1, computational basis), D(ρ1, plus/minus basis), E (ρ2, computational basis), and F (ρ2, plus/minus ba-sis). B: Dynamics of the relative entropy of coherence in the prepared states,along with the lower bound inferred from the evolution of their spin-spin cor-relation functions. The experimental errors are estimated as in Fig. 1.

tum states, automatically preserving their entropy and otherinformational properties. In our case, the states are insteaddegraded with time, but only their coherence in the consid-ered reference basis remains unaffected. We verified this bymeasuring other indicators of correlations [28] in our statesas a function of time. Fig. 1I shows the dynamics of entan-glement, classical, quantum, and total correlations (defined in[38]), as well as coherence. While entanglement is found toundergo a sudden death [35, 51] at ≈ 0.21 s, a sharp transi-tion between the decay of classical and quantum correlationsis observed at the switch time t? =

T H2 TC

2T H

2 +TC2

ln∣∣∣∣ c1(0)

c3(0)

∣∣∣∣ ≈ 0.043s. Such a puzzling feature has been reported earlier theoreti-cally [34, 52] and experimentally [47, 53, 54], but here we re-veal the prominent role played by coherence in this dynamicalpicture. Namely, coherence in the plus/minus basis is foundequal to quantum correlations before t? and to classical onesafter t?, thus remaining constant at all times. This novel in-terplay between coherence and correlations, observed in ournatural decohering conditions, is expected to manifest for anyvalid choice of geometric quantifiers used to measure the in-volved quantities [25]; for instance, in Fig. 1I we picked allmeasures based on relative entropy.

One might wonder how general the reported phenomena areif the initial states differ from the BD states of Eq. (2). In [38]we prove that, given an arbitrary state ρ with spin-spin corre-lation functions {c j}, its coherence with respect to any basis isalways larger than the coherence of the generalised BD statedefined by the same correlation functions. This entails that,even if coherence in arbitrary states may decay under noise,it will stay above a threshold guaranteed by the coherenceof corresponding BD states. To demonstrate this, we mod-ified our preparation scheme to engineer more general two-qubit states (Fig. 2A). We prepared two different pseudo-purestates ρ1 and ρ2, both with matching initial correlation triplec1(0) = 0.95, c2(0) = 0.62, c3(0) = −0.65, within the exper-imental accuracy. We then measured their coherence dynam-ics under natural evolution as before. For both of them, weobserved a decay of coherence (albeit with different rates) to-wards a common time-invariant lower bound, which was de-termined solely by the evolution of the spin-spin correlationfunctions, regardless of the specifics of the states (Fig. 2B).

Finally, we investigated experimentally the resilience of co-herence in a larger system, composed of four logical qubits.To this aim, we performed a more advanced NMR demon-stration in a BRUKER AVIII 600 MHz spectrometer equippedwith a prototype 6-channel probe head, allowing full and inde-pendent control of up to 5 different nuclear spins [30, 32]. Weused the 13CO-15N-diethyl-(dimethylcarbamoyl)fluoromethyl-phosphonate compound [32], whose coupling topology isshown in Fig. 3A. This molecule contains 5 NMR-active spins(1H, 19F, 13C, 31P and 15N), therefore we chose to decou-ple 15N and encode our four-qubit system in the remainingspins. Each pair of spins k, l = {H, F,C, P} were coupled toeach other by suitable scalar constants Jkl [38]. We employedan ‘Insensitive Nuclei Enhanced by Polarization Transfer’(INEPT)-like procedure [55] to prepare a generalised BD stateρ(0) = 1

16(I⊗4 + c1(0)σ⊗4

1 + c2(0)σ⊗42 + c3(0)σ⊗4

3)

with initialcorrelation functions c1(0) = 1, c2(0) = c3(0) = 0.7 [38], asdetailed in Fig. 3A. After evolution in a natural phase damping

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0.2

0.4

0.6

0.8

1.0

time (s)

quantumcoherence

C

○

◇

relative entropy

trace distance

FIG. 3. A: Pulse sequence (with time flowing from left to right) to prepare four-qubit generalised Bell states encoded in the 1H, 19F, 13C, and 31P nuclearspins of the 13CO-15N-diethyl-(dimethylcarbamoyl)fluoromethyl-phosphonate molecule (whose coupling topology is illustrated as an inset) by an INEPT-likeprocedure, where dkl = 1/(4Jkl) and Jkl is the scalar coupling between spins k and l. Light-gray rectangles denote continuous-wave pulses, used to decouplethe 15N nucleus. The dark grey bar denotes a variable pulse, applied to set the desired correlation triple {c j(0)}. Thicker (red) and thinner (blue) bars denote πand π/2 pulses, respectively; the phases of the striped π/2 pulses were cycled to construct each density matrix element. After the preparation stage, the systemwas left to decohere in its environment; π pulses were applied in the middle of the evolution to avoid Jkl oscillations. The final π/2 pulses served to produce adetectable NMR signal in the 1H spin channel. B: Evolution of (absolute values of) the correlation functions |c j | in the experimental states (points), along withtheoretical predictions (solid lines) based on phase damping noise with an effective relaxation time T2 ≈ 0.04 s. C: Experimental observation of time-invariantcoherence (in the plus/minus basis) in the four-qubit ensemble, measured by relative entropy (red circles) and normalised trace distance (green diamonds), alongwith theoretical predictions (solid lines). In panels B–C, experimental errors due to pulse imperfections and coupling instabilities result in error bars within thesize of the data points.

environment as before, the coherence dynamics was measuredby a non-tomographic detection method similarly to what wasdone in [56], reading out the correlation triple (see Fig. 3B)from local spin observables on the 1H nucleus, whose spec-trum exhibited the best signal-to-noise ratio [38]. The resultsin Fig. 3C demonstrate, albeit with a less spectacular accu-racy than the two-qubit case, time-invariant coherence in theplus/minus basis in our four-qubit complex, as measured bynormalised trace distance and relative entropy of coherence;the latter quantity also coincides with the global discord, ameasure of multipartite quantum correlations [57, 58], in gen-eralised BD states.

In conclusion, we demonstrated experimentally in two dif-ferent room temperature NMR setups that coherence, thequintessential signature of quantum mechanics [3], can re-sist decoherence under particular dynamical conditions, inprinciple with no need for external control. While only cer-tain states feature exactly time-invariant coherence in the-ory, more general states were shown to maintain a guaran-teed amount of coherence within the experimental timescales.These phenomena, here observed for two- and four-qubit en-sembles, are predicted to occur in larger systems composed byan arbitrary (even) number of qubits [25]. It is intriguing towonder whether biological systems such as light-harvestingcomplexes, in which quantum coherence effects persist un-der exposure to dephasing environments [16–18], might haveevolved towards exploiting natural mechanisms for coherenceprotection similar to the ideal one reported here; this is a topicfor further investigation [15].

While this Letter realises a proof-of-principle demonstra-tion, our findings can impact on practical applications, specif-ically on noisy quantum and nanoscale technologies. In par-

ticular, in quantum metrology [7], coherence in the plus/minusbasis is a resource for precise estimation of frequencies ormagnetic fields generated by a Hamiltonian aligned along thespin-x direction. When decoherence with a preferred transver-sal direction (e.g., phase damping noise) affects the estima-tion, as in atomic magnetometry [36, 59], a quantum en-hancement can be achieved by optimising the evolution time[36, 60] or using error correcting techniques [61, 62]. Herewe observed instances in which coherence is basically unaf-fected by transversal dephasing noise. This suggests that thestates prepared here (or others in which similar phenomenaoccur, such as GHZ states; see also [63]) could be used asmetrological probes with sensitivity immune to decoherence.Furthermore, it has been recently shown that the quantum ad-vantage in discriminating phase shifts generated by local spin-x Hamiltonians is given exactly by the ‘robustness of coher-ence’ [13] in the plus/minus basis, a measure equal to the tracedistance of coherence for BD states: this implies that the per-formance of such an operational task can in principle run un-perturbed, if the probes are initialised as in our demonstration,in presence of a natural dephasing environment. We will ex-plore these applications experimentally in future works.

ACKNOWLEDGMENTS

This work was supported by the European Research Coun-cil (Starting Grant No. 637352 GQCOP), the Brazilian fund-ing agencies FAPERJ, CNPq (PDE Grant No. 236749/2012-9), and CAPES (Pesquisador Visitante Especial-GrantNo. 108/2012), the Brazilian National Institute of Science andTechnology of Quantum Information (INCT/IQ), and the Ger-man DFG SPP 1601 (Grant No. Gl 203/7-2).

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6

SUPPLEMENTAL MATERIAL

Appendix A: Experimental details

1. Two Qubits System

a. NMR setup

The NMR experiments for two-qubit systems were per-formed in a Chloroform (CHCl3) sample enriched with 13C,prepared as a mixture of 100 mg of 99% 13C-labelled CHCl3in 0.7 mL of 99.8% CDCl3. The 1H and 13C spin-1/2 nucleiwere associated to the first and second qubits, respectively.This system is described by the Hamiltonian

H = ~ωH IHz + ~ωC IC

z + 2π~JIHz IC

z , (A1)

where Ikz is the z component of the spin angular momentum

of each nucleus k = H,C, and ωk is their Larmour frequency.On a Varian 500 MHz liquid-NMR spectrometer, where theexperiments were implemented, it corresponds to ωH/2π ≈500MHz and ωC/2π ≈ 125MHz, where J represents the weakscalar spin-spin coupling which was measured at ≈ 215 Hz.

The experiments were performed at room temperature, sothat the high temperature approximation (kBT � ~ωL) guar-antees that the thermal equilibrium density operator related tothis Hamiltonian, ρ0 = e−H/kBT /Z, can be simplified to

ρ0 ≈14

(IA ⊗ IB + ε∆ρ

), (A2)

where Z =∑

m e−Em/kBT is the associated partition func-tion, ∆ρ = IH

z + ICz /4 is the so-called deviation matrix, and

ε = ~ωH/kBT ∼ 10−5. The application of radiofrequency (rf)pulses and the evolution under spin interactions allow for easymanipulation of the thermal state ρ0 in order to produce differ-ent states with an excellent control of angle and phase. Thisprocedure is described by the Hamiltonian

H = ~(ωH − ωHr f )I

Hz + ~(ωC − ω

Cr f )I

Cz (A3)

−~ωH1 (IH

x cos φH + IHy sin φH) − ~ωC

1 (ICx cos φC + IC

y sin φC)

where ωkr f is the frequency of the rf field for nucleus k (on

resonance: ωkr f ≈ ωk), ωk

1 is the nutation frequency and φk

their respective phase. As only the deviation matrix ∆ρ isaffected by those unitary transformations, it is convenient towrite the resulting state as ρtotal =

[(1 − ε)(IA ⊗ IB) + ερ

]/4,

from which we define the logical NMR density matrix as thestate ρ ≡

[(IA ⊗ IB) + ∆ρ

]/4. The state preparation procedure

discussed in the main text refers to engineering ρ into any de-sired two-qubit BD state.

b. Decoherence processes

NMR naturally provides well characterised environments,characterised by the Phase Damping (PD) and GeneralisedAmplitude Damping (GAD) channels acting on each qubit[24]. PD is associated to loss of coherence (in the compu-tational basis) with no energy exchange and is specified by

the following Kraus operators,

KP0 =

√1 −

q(t)2I, KP

3 =

√q(t)2

σ3, (A4)

where the q(t) damping function is related to the characteristicrelaxation time T2 by q(t) = (1 − e−t/T2 ).

On the other hand, the GAD channel is associated to energyexchange between system and environment and can be writtenin Kraus operator form by

KG0 =

√p(

1 00√

1 − u(t)

), KG

1 =√

p(

0√

u(t)0 0

),

KG2 =

√1 − p

( √1 − u(t) 0

0 1

), KG

3 =√

1 − p(

0 0√

u(t) 0

),

(A5)

where u(t) = 1 − e−t/T1 and p = 1/2 − α with α = ~ωL/2kBT .As shown in Fig. 1A of the main text, during the evolution

period no refocusing pulses were applied. This implies thatthe phase damping function q(t) decayed naturally accordingto the characteristic relaxation time T2. We note that the ef-fective T2 for our experiment depends not only on the ther-mally induced fluctuations of longitudinal fields (standard so-called T ∗2 NMR contribution) but also on static field inhomo-geneities. This dependence makes the PD decoherence pro-cess occur faster, guaranteeing that no GAD effects should beeffectively observed during the experiment time domain. Thecharacteristic relaxation times were measured as T H

1 = 7.53s, T H

2 = 0.14 s, TC1 = 12.46 s, TC

2 = 0.90 s, which satisfyT1 � T2, as desired.

c. Quantum state tomography

The quantum state tomography for this two-qubit systemwas performed applying the simplified procedure proposed inRef. [48]. In this case the full matrix reconstruction is ob-tained after performing the local operations: II, IX, IY , XXon each qubit. Here, I, X and Y , correspond, respectively, tothe identity operation, a π/2 rotation around the x-axis, anda π/2 rotation around the y-axis. This set of operations pro-vides a 16 × 16 system of equations, whose solution gives thedensity matrix elements.

2. Four Qubits System

a. NMR setup

The four-qubit experiment was performed in a BRUKERAVIII 600 MHz spectrometer equipped with a proto-type 6-channel probe head as described in the maintext, see [32] for further details. The molecule cho-sen to encode the 4-qubit spin system was the 13CO-15N-diethyl-(dimethylcarbamoyl)fluoromethyl-phosphonate com-pound, whose coupling topology is shown in Fig. 3A; a de-tailed description of its synthesis can be found again in [32].This molecule contains 5 NMR-active spins (1H, 19F, 13C, 31P

7

and 15N). After decoupling 15N, the Hamiltonian of the re-maining four-spin system can be written as

H = ~(ωH − ωHr f )I

Hz + ~(ωF − ω

Fr f )I

Fz + ~(ωC − ω

Cr f )I

Cz +

+ ~(ωP − ωPr f )I

Pz + 2π~JHC IH

z ICz + 2π~JHF IH

z IFz + (A6)

+ 2π~JHPIHz IP

z + 2π~JFC IFz IC

z + 2π~JFPIFz IP

z + 2π~JPC IPz IC

z ,

where ωk is the Larmour frequency of each spin nucleus, mea-sured in ωH/2π ≈ 600 MHz, ωF/2π ≈ 565 MHz, ωC/2π ≈151 MHz, ωP/2π ≈ 243 MHz. The term Jkl represents thescalar spin-spin coupling constant between spins k and l. Inthis experiment, all density operator components were gen-erated by the 1H magnetisation using the interactions of 1Hwith the other nuclei, therefore the relevant Jkl couplings weremeasured, obtaining: JHF ≈ 46.45 Hz, JHP ≈ 9.64 Hz,JHC ≈ 3.99 Hz. As 1H presented the spectrum with the bestsignal-to-noise ratio, all experiments were also recorded in the1H channel. The experimentally determined transverse relax-ation times [32] were 6.2 s (1H), 15.4 s (13C), 4.6 s (19F) and0.2 s (31P). The effective T2 of our observed dynamics wasestimated at 0.04 s.

The initialM34 (alias generalised BD) state,

ρ(0) =1

16

(I⊗4 + c1(0)σ⊗4

1 + c2(0)σ⊗42 + c3(0)σ⊗4

3

), (A7)

was prepared according to the pulse sequence displayed inFig. 3A in the main text. Specifically, to distinguish the phys-ical qubits, we associate each σ⊗4

i term to the spin operatorsIHi IF

i ICi IP

i . Each of these terms is independent of the othersand also interacts independently with the environment (con-sidering the same kind of decoherence process described be-fore for the two-qubits system), therefore each term was pre-pared separately.

Firstly, a continuous-wave (cw) pulse was applied in 19F,13C, and 31P to guarantee that all terms would be generatedfrom the 1H magnetisation. Then, a variable pulse was ap-plied to 1H in order to produce the correct scaling, i.e. thecorrelation triple (c1(0) = 1, c2(0) = 0.7, c3(0) = 0.7).Each four-qubit term σ⊗4

i was then achieved by an INEPT-like transfer step. After this block, a π/2 pulse was applied onthe first spin (1H) to produce the σ⊗4

z term and/or on the other3 spins to produce σ⊗4

x /σ⊗4y , where the pulse phase was appro-

priately changed. To guarantee the quality of the final state, aphase cycling was introduced in this step. It was designed topreserve the four-qubit coherence term σ⊗4

i and eliminate theothers. This was accomplished with an 8-step phase cyclingcorresponding to a 45◦ step rotation.

b. Evolution and acquisition

After the state preparation stage, the system was allowedto interact with its natural environment and suffer decoher-ence in the form of phase damping. During this period, the Jklcouplings between some of our spins and 15N could interfereso a cw decoupling pulse was applied to the latter channel.A refocusing π pulse was applied to the other spins in orderto account for some inhomogeneity effects and avoid oscil-lations in the measured signal, due to any other Jkl coupling

evolutions. Notice that these inter π pulse delays were muchlonger (order of miliseconds) compared to the correlation timeof the thermal fluctuations (order of nanoseconds) of the inter-nal fields responsible for the decoherence; therefore, they didnot act as a decoupling field, and the evolution of the samplewas still effectively control-free.

The final step amounted to a π/2 pulse to convert eachmultiqubit element (like e.g. IH

x IFz IC

z IPz ), in a NMR-detectable

term on the 1H spin channel. This method is akin to whatwas implemented in [56], where it was shown that a directdetection is as good as a full state tomography to assess thedynamics of BD states under noise preserving their BD form.

Appendix B: Theoretical details

1. Geometric quantifiers of coherence and correlations

A rigorous and general formalism to quantify the coherenceof a d-dimensional quantum state ρ with respect to a givenreference basis {|ei〉}

di=1 can be found in [26] within the set-

ting of quantum resource theories [39]. Natural candidates forquantifying coherence arise from a rather intuitive geometricapproach, wherein the distance from ρ to the set of states di-agonal in the reference basis (known as incoherent states) isconsidered, provided the adopted distance satisfies the follow-ing constraints: contractivity under completely positive tracepreserving (CPTP) maps, i.e. D(Φ(ρ),Φ(τ)) ≤ D(ρ, τ) forany CPTP map Φ; joint convexity, i.e. D(

∑i piρi,

∑i piτi) ≤∑

i piD(ρi, τi) for any probability distribution {pi}; plus someadditional properties listed in [40]. In particular,

CD(ρ) ≡ infδ∈I

D(ρ, δ), (B1)

with I being the set of incoherent states, defines full coher-ence monotones CD (i.e., satisfying all requirements of the re-source theory defined in [26]) if one chooses for example thefollowing distance functionals: relative entropy distance [26]DRE(ρ, τ) = S (ρ||τ), where S (ρ||τ) = Tr[ρ(log(ρ) − log(τ))] isthe quantum relative entropy; and fidelity based distance [27]

DF(ρ, τ) = 1 − F(ρ, τ), where F(ρ, τ) =

(Tr

(√√ρτ√ρ))2

isthe Uhlmann fidelity. It is still currently unknown whether thetrace distance DTr (ρ, τ) = Tr

( √(ρ − τ)2

)induces a full coher-

ence monotone (in particular, it is still unclear whether prop-erty C2b of [26] is satisfied by such a measure), even thoughit is both contractive and jointly convex. However, for BDstates the trace distance of coherence is equal to the l1 normof coherence (note that we adopt a normalised definition forthe trace distance equal to twice the conventional one [24]),which is a full coherence monotone [25, 26].

Analogously, one can define faithful measures of correla-tions such as total correlations, discord-type quantum correla-tions and entanglement in the following way.

For total correlations,

TD(ρ) ≡ infπ∈P

D(ρ, π), (B2)

where π = ρA ⊗ τB, with ρA (τB) being an arbitrary state ofsubsystem A (B), form the set of product states P.

8

For quantum correlations [28],

QD(ρ) ≡ infχ∈C

D(ρ, χ), (B3)

where χ =∑

i j pi j|iA〉〈iA| ⊗ | jB〉〈 jB|, with {pi j} being a jointprobability distribution and {|iA〉} ({| jB〉}) an orthonormal basisof subsystem A (B), form the set of classical states C.

For entanglement [40],

ED(ρ) ≡ infσ∈S

D(ρ, σ), (B4)

where σ =∑

i piρAi ⊗ τ

Bi , with {pi} being a probability distri-

bution and ρAi (τB

i ) arbitrary states of subsystem A (B), formthe set of separable states S.

Finally, within this unifying distance-based approach, yet ina quite different way, it is also possible to identify the classicalcorrelations of a state ρ as follows:

PD(ρ) = inf{χρ:QD(ρ)=D(ρ,χρ)}

infπ∈P

D(χρ, π), (B5)

where the first infimum is taken with respect to the subset ofC formed by all the classical states χρ which solve the opti-misation in Eq. (B3) (i.e., all the classical states which are theclosest to ρ according to D), and the second infimum corre-sponds to the distance between each such χρ and the set ofproduct states P [41–43].

2. Conditions for time-invariant coherence

a. Two qubits

We now outline the general conditions such that constantcoherence can be observed for all time [25], constant quan-tum correlations can be observed up to a switch time t? [52],and constant classical correlations can be observed after theswitch time t?, when considering two-qubit BD states under-going local nondissipative decoherence.

Consider two qubits A and B initially in a BD state:

ρ(0) =14

IA ⊗ IB +

3∑α=1

cα(0)σAα ⊗ σ

Bα

(B6)

and such that the following special initial condition is satis-fied,

ci(0) = −c j(0)ck(0), (B7)

where I is the 2 × 2 identity, σα is the α-th Pauli matrix and{i, j, k} is a fixed chosen permutation of {1, 2, 3}. Let the two(noninteracting) qubits undergo local Markovian flip-type de-coherence channels towards the k-th spin direction (i.e. bit-flipnoise for k = 1, bit-phase-flip noise for k = 2, and phase-flipnoise for k = 3). Their evolved global state at any time t isthen represented by

ρ(t) =

3∑α,β=0

Kα ⊗ Kβρ(0)K†α ⊗ K†β , (B8)

where

K0 =

√1 −

q(t)2I, Ki = 0, K j = 0, Kk =

√q(t)2σk, (B9)

q(t) = 1 − e−γt is the strength of the noise, γ is the decoher-ence rate, and {i, j, k} is the permutation of {1, 2, 3} fixed in theinitial condition, Eq. (B7). One can easily see that the evolvedstate is still a BD state, whose corresponding correlation func-tion triple is given by

ci(t) = ci(0)e−2γt, c j(t) = c j(0)e−2γt, ck(t) = ck(0). (B10)

Focusing first on coherence, in [25] it was shown that, ac-cording to any contractive and jointly convex distance, oneof the closest incoherent states δ(α)

ρ(t) to the evolved BD stateρ(t), with respect to the basis consisting of tensor productsof eigenstates of σα, is just the Euclidean projection of ρ(t)onto the cα-axis, i.e. it is still a BD state and its triple isgiven by {δαβcα(t)}3β=1, for any α , i. Moreover, in [42] it wasshown that any contractive distance between the evolved BDstate ρ(t) and its Euclidean projection onto the c j-axis mustbe constant for any t. It immediately follows that any validdistance-based measure of coherence C( j)

D (ρ(t)) of the evolvedstate, with respect to the product basis consisting of tensorproducts of eigenstates of σ j, is invariant for any time t.

For quantum correlations, according to any contractive andjointly convex distance, one of the closest classical states χρ(t)to the evolved BD state ρ(t) is just the Euclidean projection ofρ(t) onto the closest c-axis, with triple given by {δαβcα(t)}3β=1

and α set by |cα(t)| = max{|cβ(t)|}3β=1 [52]. When |c j(0)| >

|ck(0)| then α = j until the switch time t? = 12γ ln

∣∣∣∣ c j(0)ck(0)

∣∣∣∣, withα = k afterwards. Combined with the result in [42] that anycontractive distance between the evolved BD state ρ(t) and itsEuclidean projection onto the c j-axis must be constant for anyt, this immediately implies time-invariance of quantum cor-relations up until the switch time t?. No general proof hasyet been found for the subsequent time-invariance of classicalcorrelations after the switch time t? for any contractive andjointly convex distance, but this has been observed in partic-ular cases (based e.g. on relative entropy, trace and fidelitybased distances) [34, 42, 44]. The time-invariance of classi-cal correlations is related to the finite-time emergence of thepointer basis during the dynamics [45, 47].

We note that in our two-qubit experiment we have imple-mented exactly an instance of the above conditions, specifi-cally in the case i = 2, j = 1 and k = 3; the correspondingphase-flip noise reduces precisely to the PD channel occurringin NMR, as it can be seen by comparing the Kraus operators inEqs. (A4) and (B8). The decoherence rate in our demonstra-tion was given by γ =

T H2 +TC

22T H

2 TC2

. With this evolution, the switch

time is obtained as t? = 12γ ln

∣∣∣∣ c1(0)c3(0)

∣∣∣∣. For an initial BD statewith c1(0) = 1, c2(0) = 0.7 , c3(0) = −0.7 as we prepared, re-specting the constraint in Eq. (B7), the expected switch timewas t? ≈ 0.043 s, which was found in excellent agreementwith the experimental data. Time-invariant coherence in theplus/minus basis (i.e. the eigenbasis of σ1) was observed ac-cording to any known valid geometric measure CD (Fig. 1 of

9

the main text). Entanglement and total correlations instead de-cay monotonically without experiencing any interval of time-invariance in the considered dynamical conditions.

b. Even N qubits

BD states are particular instances of a more general classof N-qubit states, that we may call generalised BD states orM3

N states [25], having all maximally mixed marginals andcharacterised only by the three correlations functions cα =

〈σ⊗Nα 〉, with j = 1, 2, 3. For any even N ≥ 2, consider N

qubits initially in anM3N state,

ρ(0) =1

2N

I⊗N +

3∑α=1

cα(0)σ⊗Nα

, (B11)

and such that the following initial condition is satisfied,

ci(0) = (−1)N/2c j(0)ck(0), (B12)

where {i, j, k} is a fixed chosen permutation of {1, 2, 3}. Letthe N (noninteracting) qubits undergo local Markovian flip-type decoherence channels towards the k-th spin direction asbefore (notice that such a dynamics is strictly incoherent [26,37] with respect to any product basis {|m〉}⊗N , with {|m〉} beingthe eigenbasis of any of the three canonical Pauli operators σmon each qubit). The evolved global state of the N qubits at anytime t is then represented by

ρ(t) =

3∑α1,...,αN =0

(Kα1 (t)⊗ . . .⊗KαN (t)

)ρ(0)

(Kα1 (t)⊗ . . .⊗KαN (t)

)†.

(B13)Once again, the evolved state is still a M3

N state, whose cor-responding correlation function triple is given by Eq. (B10).Then, for any even N ≥ 2, any valid distance-based measureof coherence C( j)

D (ρ(t)) of the evolved state, with respect to theproduct basis consisting of tensor products of eigenstates ofσ j, is invariant for any time t [25].

We note that in our four-qubit experiment (N = 4) we haveimplemented precisely an instance of the above conditions,specifically in the case i = 2, j = 1 and k = 3, and for aninitially prepared M3

4 state with c1(0) = 1, c2(0) = 0.7, andc3(0) = 0.7, respecting the constraint in Eq. (B12). Time-invariant coherence in the plus/minus basis (i.e. the eigenbasisof σ1) was observed according to various geometric measuresCD (Fig. 3 of the main text).

3. Coherence lower bound from generalised BD states

We now prove that the coherence of an arbitrary N-qubitstate ρ, with correlation functions c j = 〈σ⊗N

j 〉, is lowerbounded by the coherence of theM3

N state defined by the samecorrelation functions. This holds regardless of the number ofqubits N and when considering the basis consisting of tensorproducts of eigenstates of σ j, for any j = 1, 2, 3 (e.g. theplus/minus basis when j = 1, as demonstrated experimentallyin Fig. 2 of the main text for N = 2). Such a proof relies onthe following two results.

First, as shown in [46], any N-qubit state ρ can be trans-formed into anM3

N state with the same correlation functionsc j = 〈σ⊗N

j 〉 through the map Θ defined as follows:

Θ(ρ) =1

22(N−1)

∑22(N−1)

j=1U′j%U′†j (B14)

where U′j are the following single-qubit local unitaries

{U′j}22(N−1)

j=1 ={I⊗N , {U j1 }

2(N−1)j1=1 , {U j2 U j1 }

2(N−1)j2> j1=1, · · · (B15)

· · · {U j2(N−1) . . .U j2 U j1 }2(N−1)j2(N−1)>...> j2> j1=1

},

with

{U j}2(N−1)j=1 =

{(σ1 ⊗ σ1 ⊗ I⊗N−2), (I ⊗ σ1 ⊗ σ1 ⊗ I⊗N−3), . . . ,

(I⊗N−3 ⊗ σ1 ⊗ σ1 ⊗ I), (I⊗N−2 ⊗ σ1 ⊗ σ1),(σ2 ⊗ σ2 ⊗ I⊗N−2), (I ⊗ σ2 ⊗ σ2 ⊗ I⊗N−3), . . . ,

(I⊗N−3 ⊗ σ2 ⊗ σ2 ⊗ I), (I⊗N−2 ⊗ σ2 ⊗ σ2)}.

(B16)

Second, as shown below, the map Θ is an incoherent oper-ation with respect to the basis consisting of tensor products ofeigenstates of σ j, for any j = 1, 2, 3. An incoherent operationis a CPTP map that cannot create coherence, i.e. with Krausoperators {Ki} satisfying KiIK†i ⊂ I for all i, with I the setof incoherent states with respect to the chosen reference basis.Since any coherence monotone must be non-increasing underincoherent operations [26], it follows that the coherence ofΘ(ρ) is less than or equal to the corresponding coherence of ρ.

In what follows we will prove that Θ is an incoherentoperation with respect to the basis consisting of tensorproducts of eigenstates of σ1 (i.e. the plus/minus basis),although analogous proofs hold when considering the othertwo Pauli operators. Since the Kraus operators of the map Θ

are given by K j = 12N−1 U′j, in order for Θ to be an incoherent

operation in the plus/minus basis we need that U′jδU′†

j ∈ I forany δ ∈ I and any j ∈ {1, · · · , 22(N−1)}, where I is the set ofstates diagonal in this basis. This obviously holds for j = 1,being U′1 the identity. On the other hand, as it can be seenfrom Eq. (B15), all the other single-qubit unitaries U′j are justproducts of the single-qubit unitaries U j listed in Eq. (B16),so that we just need to prove that U jδU

†

j ∈ I for any δ ∈ Iand for any j ∈ {1, · · · , 2(N − 1)}. For any j ∈ {1, · · · ,N − 1},U j just leaves any state which is diagonal in the plus/minusbasis invariant, it being a tensor product between two σ1’sacting on two neighbouring qubits and the identity acting onthe remaining ones. Otherwise, for any j ∈ {N, · · · , 2(N − 1)},U j is the tensor product between two σ2’s acting on twoneighbouring qubits and the identity on the rest of the qubits.Consequently, by using σ2|±〉〈±|σ2 = |∓〉〈∓| and the fact thatthe general form of a state δ diagonal in the plus/minus basisis δ =

∑j1, j2,··· , jN =± p j1, j2,··· , jN | j1, j2, · · · , jN〉〈 j1, j2, · · · , jN |,

we have that, when e.g. j = N, then UNδU†

N =∑j1, j2,···, jN =± p j1, j2,···, jN UN | j1, j2, · · ·, jN〉〈 j1, j2, · · ·, jN |U

†

N =∑j1, j2,···, jN =± p j1, j2,···, jN |π( j1), π( j2), · · ·, jN〉〈π( j1), π( j2), · · ·, jN |,

where π(±) ≡ ∓, so that UNδU†

N ∈ I. Analogously, one cansee that all the remaining single-qubit local unitaries U j aresuch that U jδU j ∈ I, thus completing the proof.